3.1597 \(\int \frac{(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{2 e (b d-a e)}{3 b^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^2}{4 b^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^2}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

-(b*d - a*e)^2/(4*b^3*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*e*(b*d - a
*e))/(3*b^3*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - e^2/(2*b^3*(a + b*x)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.193057, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{2 e (b d-a e)}{3 b^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^2}{4 b^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^2}{2 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

-(b*d - a*e)^2/(4*b^3*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*e*(b*d - a
*e))/(3*b^3*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - e^2/(2*b^3*(a + b*x)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 11.1123, size = 82, normalized size = 0.66 \[ \frac{e \left (d + e x\right )^{3}}{12 \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{3}}{8 \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

e*(d + e*x)**3/(12*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) + (2*a +
2*b*x)*(d + e*x)**3/(8*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2))

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Mathematica [A]  time = 0.0579588, size = 69, normalized size = 0.55 \[ \frac{-a^2 e^2-2 a b e (d+2 e x)+b^2 \left (-\left (3 d^2+8 d e x+6 e^2 x^2\right )\right )}{12 b^3 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-(a^2*e^2) - 2*a*b*e*(d + 2*e*x) - b^2*(3*d^2 + 8*d*e*x + 6*e^2*x^2))/(12*b^3*(
a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.01, size = 69, normalized size = 0.6 \[ -{\frac{ \left ( bx+a \right ) \left ( 6\,{x}^{2}{b}^{2}{e}^{2}+4\,xab{e}^{2}+8\,x{b}^{2}de+{a}^{2}{e}^{2}+2\,abde+3\,{b}^{2}{d}^{2} \right ) }{12\,{b}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/12*(b*x+a)/b^3*(6*b^2*e^2*x^2+4*a*b*e^2*x+8*b^2*d*e*x+a^2*e^2+2*a*b*d*e+3*b^2
*d^2)/((b*x+a)^2)^(5/2)

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Maxima [A]  time = 0.756954, size = 178, normalized size = 1.42 \[ -\frac{2 \, d e}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{a^{2} b^{2} e^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \, a b e^{2}}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{d^{2}}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{a d e}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

-2/3*d*e/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 1/4*a^2*b^2*e^2/((b^2)^(9/2)*(x
+ a/b)^4) + 2/3*a*b*e^2/((b^2)^(7/2)*(x + a/b)^3) - 1/2*e^2/((b^2)^(5/2)*(x + a
/b)^2) - 1/4*d^2/((b^2)^(5/2)*(x + a/b)^4) + 1/2*a*d*e/((b^2)^(5/2)*b*(x + a/b)^
4)

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Fricas [A]  time = 0.205817, size = 132, normalized size = 1.06 \[ -\frac{6 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 2 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/12*(6*b^2*e^2*x^2 + 3*b^2*d^2 + 2*a*b*d*e + a^2*e^2 + 4*(2*b^2*d*e + a*b*e^2)
*x)/(b^7*x^4 + 4*a*b^6*x^3 + 6*a^2*b^5*x^2 + 4*a^3*b^4*x + a^4*b^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((d + e*x)**2/((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.599493, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x